A blind spot in undergraduate mathematics: The circular definition of the length of the circle, and how it can be turned into an enlightening example

Alexei Vernitski

We highlight the fact that in undergraduate calculus, the number pi is
defined via the length of the circle, the length of the circle is defined as a
certain value of an inverse trigonometric function, and this value is defined
via pi, thus forming a circular definition. We present a way in which this
error can be rectified. We explain that this error is instructive and can be
used as an enlightening topic for discussing different approaches to
mathematics with undergraduate students.